7.1 Decomposition of primes in extensions

Prime decomposition in number fields reveals how rational primes behave when extended to larger algebraic structures. This topic explores how primes can split, remain inert, or ramify, providing crucial insights into the arithmetic of number fields.

Understanding prime decomposition is key to grasping fundamental concepts in algebraic number theory. It connects to broader ideas like ideal factorization, ramification theory, and the structure of Dedekind domains, forming a foundation for more advanced studies in the field.

Prime Ideals in Number Fields

Concept and Properties of Prime Ideals

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• Prime ideals in number fields represent proper ideals that cannot be further factored into smaller ideals, analogous to prime numbers in the integers
• Ring of integers of a number field forms a Dedekind domain ensuring unique factorization of every nonzero ideal as a product of prime ideals
• Prime ideals generalize the concept of prime numbers allowing for a more refined study of divisibility in algebraic number theory
• Norm of a prime ideal always equals a power of a rational prime ($p^f$ where $f$ is the inertia degree) measuring the "size" of the prime ideal
• Factorization of ideals in number fields provides information about rational prime decomposition when extended to the number field
  • Example: In $\mathbb{Q}(\sqrt{-5})$, the ideal $(2)$ factors as $(2, 1 + \sqrt{-5})(2, 1 - \sqrt{-5})$
• Concept of prime ideal factorization crucial for understanding ramification and splitting behavior of primes in field extensions
  • Example: In a quadratic field $\mathbb{Q}(\sqrt{d})$, a prime $p$ can split, remain inert, or ramify depending on whether $d$ is a quadratic residue modulo $p$

Applications and Significance

• Prime ideals form the building blocks for the unique factorization of ideals in number fields
• Study of prime ideals leads to important number-theoretic results such as the Chebotarev density theorem
• Prime ideals play a crucial role in the theory of Dedekind zeta functions and L-functions
• Understanding prime ideals essential for solving Diophantine equations over number fields
• Prime ideals provide a framework for generalizing results from elementary number theory to algebraic number fields
  • Example: Generalizing the law of quadratic reciprocity to higher degree extensions

Decomposition of Rational Primes

Characterization of Prime Decomposition

• Decomposition of a rational prime $p$ in a number field $K$ described by factorization of principal ideal $(p)$ into prime ideals in the ring of integers of $K$
• Decomposition type of a prime characterized by degrees and multiplicities of prime ideals in its factorization
  • Example: In $\mathbb{Q}(\sqrt{-5})$, $(2) = \mathfrak{p}_1\mathfrak{p}_2$ where $\mathfrak{p}_1 = (2, 1 + \sqrt{-5})$ and $\mathfrak{p}_2 = (2, 1 - \sqrt{-5})$
• Eisenstein's criterion and its generalizations used to determine irreducibility of polynomials often serving as the first step in analyzing prime decomposition
  • Example: $x^3 + 3x^2 + 3x + 3$ is Eisenstein with respect to $p=3$, thus irreducible over $\mathbb{Q}$
• Decomposition field of a prime $p$ defined as the fixed field of the decomposition group providing information about the splitting behavior of $p$
• Local methods such as $p$-adic analysis employed to study prime decomposition in specific cases
  • Example: Using Hensel's lemma to lift factorizations from finite fields to $p$-adic fields

Specific Cases and Applications

• Decomposition of primes in cyclotomic fields follows specific patterns related to congruence conditions modulo the conductor of the field
  • Example: In $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive $p$-th root of unity, primes $q \neq p$ split completely if and only if $q \equiv 1 \pmod{p}$
• Understanding prime decomposition essential for computing important invariants like the discriminant and different of number fields
• Prime decomposition in quadratic fields determined by the Legendre symbol and quadratic reciprocity
• Decomposition behavior in Galois extensions uniform for all prime ideals lying above a given rational prime
• Study of prime decomposition crucial for understanding the arithmetic of elliptic curves and modular forms over number fields

Kummer-Dedekind Theorem for Factorization

Statement and Application of the Theorem

• Kummer-Dedekind theorem provides a method for factoring prime ideals in number fields defined by irreducible polynomials
• Theorem relates factorization of a prime $p$ to factorization of a defining polynomial modulo $p$ in the polynomial ring over the finite field $\mathbb{F}_p$
• Each irreducible factor of the polynomial mod $p$ corresponds to a prime ideal in the factorization with the factor's degree determining the inertia degree
  • Example: For $\mathbb{Q}(\sqrt{2})$ and $p=7$, $x^2 - 2 \equiv (x-3)(x+3) \pmod{7}$, so $(7)$ splits into two distinct prime ideals
• Multiplicity of each factor in the polynomial factorization determines the ramification index of the corresponding prime ideal
• Applying the theorem requires finding factorization of polynomials over finite fields using algorithms such as Berlekamp's algorithm
• Kummer-Dedekind theorem particularly useful for computing prime ideal factorizations in number fields of small degree or with simple defining polynomials
  • Example: In cubic fields defined by $x^3 + ax + b$, the theorem easily determines the splitting behavior of primes not dividing the discriminant

Limitations and Extensions

• In cases where the theorem doesn't directly apply such as when the ring of integers is not monogenic additional techniques like computing integral bases may be necessary
• Theorem can be generalized to handle cases where the defining polynomial is not irreducible or the field is not monogenic
• For large degree number fields computational complexity of polynomial factorization over finite fields can become a bottleneck
• Kummer-Dedekind theorem forms the basis for more advanced techniques in computational algebraic number theory
• Understanding the theorem crucial for implementing computer algebra systems for number field computations

Inert, Split, or Ramified Prime Ideals

Classification of Prime Ideals

• Prime $p$ inert in a number field $K$ if the principal ideal $(p)$ remains prime in the ring of integers of $K$
  • Example: In $\mathbb{Q}(i)$, the prime 3 is inert as $(3)$ remains prime
• Prime $p$ splits completely in $K$ if $(p)$ factors into $n$ distinct prime ideals where $n$ is the degree of $K$ over $\mathbb{Q}$
  • Example: In $\mathbb{Q}(\sqrt{5})$, the prime 11 splits as $(11) = (11, 4 + \sqrt{5})(11, 4 - \sqrt{5})$
• Prime $p$ ramified if at least one prime ideal appears with multiplicity greater than 1 in the factorization of $(p)$
  • Example: In $\mathbb{Q}(\sqrt{-3})$, the prime 3 ramifies as $(3) = (3, 1 + \sqrt{-3})^2$
• Ramification index $e$ and inertia degree $f$ of a prime ideal $\mathfrak{P}$ lying above $p$ satisfy the fundamental equality $efg = n$ where $g$ is the number of distinct prime ideals in the factorization and $n$ is the field degree
• For Galois extensions decomposition behavior uniform for all prime ideals lying above a given rational prime

Significance and Applications

• Discriminant of a number field divisible precisely by the ramified primes providing a finite set of primes to check for ramification
• Understanding the classification of primes crucial for studying more advanced topics such as ideal class groups and the distribution of prime ideals
• Classification of primes important in the study of zeta functions and L-functions of number fields
• Splitting behavior of primes determines the decomposition of primes in composite field extensions
• Inert primes play a role in determining the structure of the unit group of number fields
• Ramification theory extends to more general contexts such as extensions of local fields and geometric ramification in algebraic geometry